Grease Flow in Elbow channel

Abstract

The flow of lubricating greases in an elbow channel have been modeled
and validated with velocity profiles from flow visualizations using micro
Particle Image Velocimetry. The elbow geometry induce a non-symmetric distribution
of shear stress throughout its cross section, as well as varying shear
rates through the transition from the elbow inlet to the outlet. The flow has
been modeled both for higher flow rates and for creep flow. The influence of
the grease rheology and flow conditions to wall slip, shear banding and an
observed stick-slip type of motion observed for low flow rates are presented.
The effect on the flow of the applied pressure is also modeled showing that
the flow is sensitive to the pressure in the angular (ϕ) direction of the elbow.
For high pressures it is shown that the flow is reversed adjacent to the elbow
walls.

Full text

Noname manuscript No.
(will be inserted by the editor)
Grease flow in an elbow channel
Lars G. Westerberg ·Josep
Farr´e-Llad´os ·Jinxia Li ·Erik H¨oglund ·
Jasmina Casals-Terr´e
Received: date / Accepted: date
Abstract The flow of lubricating greases in an elbow channel have been mod-
eled and validated with velocity profiles from flow visualizations using micro
Particle Image Velocimetry. The elbow geometry induce a non-symmetric dis-
tribution of shear stress throughout its cross section, as well as varying shear
rates through the transition from the elbow inlet to the outlet. The flow has
been modeled both for higher flow rates and for creep flow. The influence of
the grease rheology and flow conditions to wall slip, shear banding and an
observed stick-slip type of motion observed for low flow rates are presented.
The effect on the flow of the applied pressure is also modeled showing that
the flow is sensitive to the pressure in the angular (ϕ) direction of the elbow.
For high pressures it is shown that the flow is reversed adjacent to the elbow
walls.
Keywords Grease flow ·micro Particle Image Velocimetry ·Herschel-Bulkley
rheology ·Analytical modeling ·Wall slip ·Shear banding ·Stick-slip flow ·
Flow separation
Lars G. Westerberg∗
Division of Fluid and Experimental Mechanics, Lule˚a University of Technology, SE-971 87
Lule˚a, Sweden
∗E-mail: lars-goran.westerberg@ltu.se
Josep Farr´e-Llad´os ·Jasmina Casals-Terr´e
Department of Mechanical Engineering, UPC - Technical University of Catalonia, Colom
7-11 08222 Terrassa (BCN), Spain.
Jinxia Li ·Erik H¨oglund
Division of Machine Elements, Lule˚a University of Technology, SE-971 87 Lule˚a, Sweden

2 L.G. Westerberg et al.
1 Introduction
Grease and oil are widely used to lubricate moving mechanical parts in various
mechanical systems such as gearboxes, engines and hub-units for trucks and
passenger cars. Grease has in many applications been found to be preferable
to oil as it due to its characteristic consistency prevent leakage and also con-
tribute to a systems sealing properties. Grease also adheres to the surfaces
which prevents corrosion and contribute to a lower friction. The main disad-
vantage with grease lubrication is the need of relubrication as the lifetime of
the machine element often exceeds the lifetime of the grease.
Grease is a multi-phase material composed of a thickener, base oil and
additives. The thickener determines the greases basic properties and forms a
network structure in which the base oil is kept through Van der Waals- and
capillary forces [4]. The thickener is also used to classify greases into different
types ([20], p.49). A typical distribution is usually 80-90 % oil and 10-20 %
thickener.
The thickener induce the grease’s characteristic visco-elastic and yield be-
haviour [13], meaning the viscosity of the grease is strongly dependent on
the shear rate (usually decreasing with increasing shear rate, so called shear
thinning ([20], p.112) and that the onset of flow is dependent on a threshold
value of the shear stress. These properties gives grease different lubrication
properties compared to oil. Oils have a Newtonian rheology which enables the
running tracks in loaded contacts to be sufficiently replenished as the flow
responds to the centrifugal forces and shear stresses induced by the motion of
the geometry. This condition is necessary in order to have an optimum lubri-
cation as the lubricant is pushed away from the contact due to the high load
(shear stresses), which in turn implies a reduced amount of available lubricant
in the running tracks. Greases however, will not without much difficulty flow
back into the contact once they initially have been pushed out. Baart et al.
[2] showed that the replenishment in the tracks of loaded contacts for grease
lubrication also is supported by oil bleeding. However, as presented by Lugt et
al. [21] oil bleeding may negatively affect the replenishment if the grease is in
the wrong place. Also, too much grease will increase the friction due to heavy
churning.
In order to understand and being able to optimize the lubrication mecha-
nism, an understanding of grease flow dynamics is important. Further, a good
lubrication is also dependent on a sufficient supply of grease into the contacts.
In lubrication systems the grease is pumped from a reservoir through pipes
connected by means of elbow- and t-junction fittings. Here the grease flow dy-
namics in such curved geometries plays an important role for the pumpability
of grease through these systems.
Studies of grease flow in pipes has been done by e.g. Mahncke and Ta-
bor [22], and by Froishterer et al. [12]. Radulescu et al. [24] investigated the
flow in pipes with discontinuities, which also could be applied to the flow in
bearings. This work is a continuation of previous studies of the flow in closed
straight channels [29,17] and in a 3D double restriction seal geometry [14, 1,

Grease flow in an elbow channel 3
18]. A channel with a 90◦elbow is used and an analytical model of the flow
is developed and validated with experimental data from micro Particle Im-
age Velocimetry (µPIV) measurements of the flow in the elbow. This work is
motivated by the value of an increased understanding of the flow motion of
lubricating grease in curved geometries, both with respect to the fundamental
scientific aspect, but also to the engineering application of lubrication systems.
The present study is also valuable as validation of numerical models.
2 Analytical model of the flow in the channel elbow
In this part an analytical model of the flow in the channel elbow is presented.
Cylindrical coordinates (r, ϕ, z) are used to describe the flow; see Fig. 1 for
a schematic view of the geometry. For the µPIV measurements presented in
§3 a channel with a rectangular cross section of 250x1000 µm is used. The
elbow section channel have an inner/outer radius of 0.875/1.125 mm. For a
corresponding model of the flow in the straight sections the reader is referred
to the work by Westerberg et al. [29]. The flow is considered incompressible
and the velocity uis considered to only have a component in the ϕ-direction
and only be dependent on the radial position, i.e. u=uϕ(r). This means that
the equation of motion solely can be treated to govern the flow. With one
unknown velocity component the continuity equation is redundant. On tensor
form the equation of motion reads
ρDui
Dt =−pi+ρFi+τij,j ,(1)
where D/Dt is the material derivative, ρthe grease density, uithe velocity
field, τij the stress tensor, pithe pressure gradient, and Fia volume force
which for the present case is gravity which in turn can be considered negligible.
Considering the definition of the material derivative the equation of motion is
on vector form written
ρ∂u
∂t +ρ(u· ∇)u=−∇p+∇ · τij .(2)
For a stationary flow the time derivative of the velocity is zero. Further, con-
sidering a 1D flow the velocity in the elbow only has a component in the
longitudinal (ϕ) direction, and the velocity is only dependent on the radial po-
sition in the elbow (u=uϕ(r)). This means that the only non-zero component
of Eq. (2) is the ϕ-component. Consequently the only non-zero component of
the shear stress tensor is τrϕ which equals τϕr as the stress tensor is symmetric
([5], §3.2). Using the definition of the Nabla operator in cylindrical coordinates
Eq. (2) then reduces to ([5], p. 612)
0 = 1
r2
∂
∂r r2τrϕ−1
r
∂p
∂ϕ .(3)

4 L.G. Westerberg et al.
For a Poiseuille flow in a straight channel the pressure distribution can be
considered to decrease linearly along the length of the channel (i.e. in the
main flow direction (x) and vary in the transverse direction (y) to the main
flow direction such that
p=αx +f(y),(4)
where αis a constant and f(y) is an arbitrary function. Here αis negative
meaning the pressure varies across a section of the channel but decreases along
the central surface. The pressure gradient in the main flow direction then reads
∇p=∂p
∂x =α. (5)
For the case of a flow in an elbow channel the pressure distribution can be
considered analogous comparing the azimuthal cylindrical coordinate ϕwith
xin the straight channel. Curved channel flow is quite a large area within the
fluid mechanical community as the curvature(s) easily for Newtonian fluids
like water induce several interesting phenomena such as flow separation and
instabilities. The work by Dean [8] is a classical paper within the field and for
the flow in the elbow in the present analysis we consider an analogous pressure
distribution
p=αϕ +f(r),(6)
where f(r) is an arbitrary function. The pressure gradient then yields (recalling
cylindrical coordinates are used in the elbow)
∇p=1
r
∂p
∂ϕ =α. (7)
From Eq. (3) it then follows that
∂
∂r r2τrϕ=−rα. (8)
Solving for τrϕ results in (neglecting the indexes for simplicity)
τ=−α
2+C1
r2,(9)
where C1is a constant. The Herschel-Bulkley rheology model is considered to
govern the relation between the shear stress and the shear rate in the grease.
In cylindrical coordinates and for the present flow condition (u=uϕ(r)) the
Herschel-Bulkley model reads [18]

Grease flow in an elbow channel 5
τ=τ0+Kr∂
∂r uϕ
rn
.(10)
Here τ0is the yield stress, Kthe consistency of the grease, and nthe power
law exponent which for a shear thinning material like grease typically is less
than one. Eq. (10) into Eq. (9) gives
Kr∂
∂r uϕ
rn
+τ0=−α
2+C1
r2,(11)
i.e.
r∂
∂r uϕ
r=1
K1
n−α
2+C1
r2−τ0
1
n
.(12)
Solving for uϕresults in
uϕ(r) = r
K1
n1
r−α
2+C1
r2−τ0
1
n
dr +C2r, (13)
which is a general solution for the velocity in the viscous region in connection
to respective boundary in the channel elbow; see Fig. 2. C2is a new constant.
The velocity profile has in the presence of wall slip three characteristic regions
in connection to each boundary, going from respective boundary towards the
middle of the elbow [29]: a very thin slip layer, a viscous region where the grease
is fully yielded, i.e. |τ|> τ0, and a plug region where the grease behaves as a
solid body |τ|< τ0. For the case of no slip and no plug flow the two viscous
layers fill the channel width. The slip layer is identified from the experiments
by extrapolating the velocity profile towards the origin representing the inner
radius of the channel, and towards opposite boundary (outer radius); cf. Fig. 1.
For a no slip condition the velocity would approach zero continuously, but as
illustrated in the figure this is not the case; rather than having a continuous
behavior of the velocity profile, the velocity gradient appears to be significantly
higher in the region between the solid boundary (inner radius) and the first
registered data point for the velocity. For a flow scenario as in Fig. 2 the slip
effect is greater close to the inner radius compared to the corresponding effect
at the opposite boundary.
For a flow in a straight channel, the velocity profile is symmetric along the
centerline of the channel; this has not to be the case in a curved channel as
the non-symmetric geometry may induce an increasing velocity in the positive
radial direction and hence induce asymmetry in the flow. The yield point is
the location(s) in the flow where the shear stress in the grease equals the yield
stress. For the case of a non-existing plug layer, the yield point then is the point
of maximum velocity; cf. a Newtonian Poiseuille flow in a straight channel
which has a characteristic parabolical velocity profile. The yield point in this

6 L.G. Westerberg et al.
case is at the channel center. For the case of a plug flow in a non-symmetric
flow as in the channel elbow, the velocity profile has two yield points: one on
each side of the centerline of the channel. These points constitute the limits of
the plug region. The analytical velocity profile across the channel elbow is then
built by the plug region (if present), and the viscous solutions in connection
to the plug region on respective side of the channel centerline; see Fig. 2.
Matching the velocity profile with the µPIV measurements, the data points
of the velocity can be used as boundary conditions to solve for the unknown
constants in Eq. (13). In channel flows a no slip condition at the channel
wall is typically used. However, if slip is present the no slip condition is no
longer valid. Following the methodology presented in Westerberg et al. [29] the
velocity data point being located closest to the channel wall is considered as
boundary condition in this paper (u1in Fig. 2). Eq. (13) can then be re-written
as
uϕ(r) = uϕ1(r) = r
K1
n
r

r1
1
r−α
2+C1
r2−τ0
1
n
dr +ru1
r1
.(14)
A general solution for an arbitrary value of ndoes not exist but for certain
fractions a solution can be found. Two fractions are n= 1/2 and n= 1/3
respectively. The greases used in this paper all have an n-value of the order
of 1; see Tab. 1. In the studies by Li et. al [18] and Westerberg et. al [29]
it is also shown that Kand τ0have larger impact on the flow as they span
over larger magnitudes. It is found that a good match between the analytical
model and the measured velocity profile is obtained for n= 1/2 for the case
of a reduced pressure and yield stress value. This behavior is likely due to the
changed rheology of the grease with a smaller n-value: as the grease viscosity
for a smaller n-value decreases faster for a given increase in shear rate the
flow separates faster (see §4.1) and the analytical model breaks down. The
n-value is henceforth considered to be 1. For n= 1 the integral in Eq. (14) is
straightforward to calculate. The solution for uϕthen reads
uϕ(r) = r
K(τ0+α) ln r1
r+C1
21
r2
1
−1
r2+ru1
r1
.(15)
To solve for C1the yield point in the measured velocity profile is identified.
With rp1as the location of the yield point and the corresponding velocity up1
the solution for C1reads
C1=−r1rp1
r2
1−r2
p1rp1r1(2τ0+α) ln rp1
r1+ 2K(up1r1−rp1u1).(16)
The solution for the velocity profile presented above constitute uϕ1(r) in Fig. 2.
The solution for uϕ2is obtained analogously considering the first measured
velocity calculated from the outer boundary, located at r=r2, and the velocity

Grease flow in an elbow channel 7
at the yield point (r=rp2). Important here is that the r-coordinate in Eq. (15)
is defined positive in the normal direction to the channel boundary pointing
into the channel.
3 Grease flow measurements using µPIV
In this section results from the flow visualizations are presented and discussed.
The µPIV measurements have been conducted in a micro-channel featuring
a 90◦elbow; see Fig. 3 for a view of the channel setup. The channel has a
rectangular cross section of 250x1000 µm. The radius at the channel centerline
is 1 mm. The channel is manufactured using a 3D printer from Stratasys,
model Objet 30. The channel is built in the polymeric material VeroWhitePlus
RGD835, and UV curing glue is used to seal the joint of the micro-channel
part and the glass. Fig. 3b shows the 3D model of the RPT part and Fig. 4 the
steps followed to manufacture the micro-channel. More details about the 3D
printing operation to build the present channels can be found in the paper by
Farr´e et al. [11]. Fig. 1 shows a schematic drawing of the micro-channel and the
locations where the µPIV measurements have been performed in the straight
section and the elbow. The exact position within the dashed regions cannot
be defined as no reference point can be introduced in the micro-channels.
However, the measurements are not performed any closer than 5 mm from the
inlet in the straight section, meaning the flow is fully developed considering
the Hagen-Poiseuille law [9].
The µPIV experimental setup is shown in Fig. 5a and the principles behind
PIV is presented in Fig. 5b. The experimental setup consists of a syringe to
inject the grease into the channel, a syringe pump to control the flow rate,
a laser to illuminate the flow, a microscope with the corresponding objec-
tive(s) and a high speed camera to capture the flow motion, and a computer
with PIV software to calculate the velocity of the flow. The main principle of
µPIV is to take a series of double frame images of the particle seeded grease
and to analyze the images in the computer to visualize the flow motion. Here
the tracer particles used are dry-powder fluorescent MF Rhodamine B parti-
cles from Microparticles GmbH (www.microparticles.de) having a diameter of
3.23 ±0.06 µm. The microscope objective has a 20x magnification. The im-
ages within each double frame image-pair are separated by a small time step
enabling the motion of the tracer particles to be calculated using a correlation
routine. Each experiment in the present study comprises the average of a total
of 500 image pairs. The system used for these experiments is a commercial
system by Dantec A/S which includes the software Dynamic Studio in which
the PIV routine is incorporated. The method of PIV is widely used to visu-
alize fluid flow, it is not limited to small scale geometries; it can theoretically
be applied to any scale as long as it is possible to record the motion of the
flow. The main difference between µPIV and ”macro” PIV is that in µPIV the
entire flow volume is illuminated by the laser light and the actual measuring
plane is set by the focal depth of the microscope, while for macro PIV a laser

8 L.G. Westerberg et al.
sheet - comprising the measuring plane - is formed using imaging optics; cf.
Fig. 5b. For more details on PIV the book by Raffel et al. [25] is recommended.
4 Results and discussion
In this section velocity profiles from flow visualizations are presented for the
grease flow in the inlet (straight section upstream of the elbow), elbow and
outlet respectively; cf. Fig. 1. The velocity profiles at the elbow have been
compared to the analytical model. Three different lithium greases with NLGI
grade 00, 1 and 2 have been used; see Tab. 1 for rheology data of the actual
greases based on the Herschel-Bukley rheology model. In order to obtain best
possible fit the values of the grease consistency Kand yield stress τ0, and the
angular pressure parameter αhave been adapted. The influence of the angular
pressure in the elbow have also been analyzed separately as it was found that
the α-value heavily determines the characteristics of the flow in the elbow. Of
specific interest are also wall slip effects, shear banding, and the transition of
the flow as the grease moves through the elbow. These aspects are analyzed
in separate subsections below.
A justified question is how correct it is to consider the rheological pa-
rameters as free parameters for the curve fit as we have a measured value
obtained from a rheometer. Important here is that rheometer measurements
not are without error sources; there are in fact many error sources related to
these measurements, see e.g. §5.2.2 in the book by Lugt [20]. Wall slip is for in-
stance not considered in rheometers and the grease is assumed to be uniformly
sheared throughout the gap. A couple of papers by the authors have shown
that these conditions in fact are violated for the case of grease flow [29,18]. A
consequence of this is the question how reliable the values from a rheometer
are in terms of grease rheology. It is hence of interest to investigate eventual
discrepancies between the rheometer values and values obtained from the fit
between the analytical model and the measured velocity profiles.
The flow rates considered in this paper range from 0.002 ml/min to 0.5 ml/min
which are the limits of the syringe pump. Higher (or lower) flow rates may be
of interest but this range is considered satisfactory at present stage. The exper-
imental results have of course a specific value on their own, but in the context
of a combined analytical model with experimental data, the latter has an im-
portant role as validation for the analytical model and for future numerical
models.
The values of the rheological parameters in the Herschel-Bulkley model
have been measured for each grease as mentioned above. Of interest is to
compare these values with the values obtained from the curve fit between
the analytical model and the µPIV data. Rheometer measurements are typ-
ically affected by errors as Lugt reports ([20], §5.2.2). For the low- to mid
Reynolds number flow covered by the measurements, the analytical model is
considered robust. For increasing Reynolds numbers the flow will eventually
become turbulent and then the model is no longer valid. By comparing the

Grease flow in an elbow channel 9
matched values of the rheology model constants with the ones obtained from
the rheometer measurements we can get an indication on the reliability of the
measured values.
4.1 Influence of the driving pressure (α-value) on the flow in the elbow
Fig. 6 shows the velocity profiles from the across the elbow as αgrows in mag-
nitude. As αincreases the trend is that the flow approaches a flow reversal
in connection to the boundaries of the elbow. Here ’reversal’ means that the
flow in connection to the curved boundaries in the elbow change direction and
hence induce an increased shear in the flow, which in turn lead to a vortical
flow (circulation) in the elbow. Within the pure scientific context the mecha-
nisms behind the reversal are of interest to build a detailed understanding of
the physics of the grease flow. Flow reversal is characterized by the negative
velocity which start to appear for α=−500 and which is fully present for
α=−1000. For a curvature radius of 1 mm these values correspond to a pres-
sure gradient in the elbow of 0.5-1 MPa/m (=5-10 bar/m); see Eq. (7). Flow
reversal is typically present in connection to flow separation initialized by an
adverse pressure gradient caused by the curved geometry [15]. Flow separation
means that the flow undergoes a transition from being detached to the channel
boundary to inherit a vortical structure. The working hypothesis here is that
flow separation is responsible for the observed flow reversal in the elbow. It
should be noted that there is a great difference between laminar separation
and turbulent separation; the latter occurring at (very) high Reynolds numbers
(see below) which should be considered exotic when it comes to grease flow
due to the high viscosity of the grease. In the limit of infinite shear rate when
the viscosity of the grease approaches the base oil viscosity turbulence may be
present in the flow if the velocity is high enough, but for the cases presented
in the present paper the flow is laminar. The core in the working hypothesis
is a combination of the elbow geometry which comprise two curvatures, and
the onset of the flow reversal as αincreases. Also, the flow reversal effect is
slightly more accentuated in connection to the inner boundary of the elbow
which has a smaller curvature radius than the outer radius hence causing a
stronger adverse pressure gradient and consequently a more pronounced sepa-
ration. Generally, in order to avoid separation, the boundary geometry should
be varied gradually in order to follow the flow streamlines; this is however not
the case in the elbow as the flow experiences a fast transition from an infinite
curvature radius (i.e. a straight channel flow) to a curvature of 1 mm. Another
interesting observation is that the critical α-value (αc) for when the flow re-
versal is initiated is highly dependent on the NLGI grade of the grease. For
NLGI 00 αc≈ −100 while for NLGI 2 αc≈ −1000 to be compared for the cor-
responding value of -500 for the NLGI 1-grease. This observation makes sense
in terms of flow separation as a less viscous fluid will have a larger Reynolds
number compared to a more viscous fluid for the same velocity. The (dimen-

10 L.G. Westerberg et al.
sionless) Reynolds number is the ratio between the inertial- and viscous forces
in the flow according to
Re=LU
η,(17)
where Lis a characteristic length scale (in channel flow typically the channel
width), Ua characteristic velocity (e.g. the mean velocity in the channel or the
maximum velocity), and ηthe kinematic viscosity which in turn is dependent
on the shear rate as lubricating greases are shear thinning materials. The
location where flow separation occur is not dependent on the Reynolds number
[15], but the overall flow characteristics - including the presence of separation
or not - is heavily influenced by the Reynolds number. The impact of the
curvature of the boundaries is also shown when comparing Fig. 6(a)-(b): in
subfigure (a) the analytical model does not fully match the measured velocity
close to the inner boundary (0 < r∗<0.3), but as the pressure is increased in
(b) the match is better. This indicates that the pressure decrease with r∗.
A natural approach to continue the present study, and also to more in
detail dig into the fundamental flow properties of lubricating greases, is to
develop the next generation test rig which enable higher flow rates and which
also induces larger gradients in the flow in order to initiate flow separation.
Flow separation is not the focus of this paper, it is a result from the developed
analytical model. But for future work it is of interest to further investigate this
phenomenon, as well as the overall vortical flow behaviour of grease. And here
an elbow channel not necessarily is the most natural geometry; to introduce a
shear layer in the flow a plate in the normal direction to a parallel flow (flow in
a straight channel) may be more effective. With respect to the application of
lubrication systems a natural future approach is to consider a flow geometry
which consists of a series of fittings in order to investigate how the flow is
developed downstream.
4.2 Shear banding
Shear banding is a phenomena in the flow induced by instabilities in the grease
phases (base oil and thickener), causing the shear rate in the flow to vary across
the channel width [10,16]. One important application where the occurrence of
shear banding have a direct impact on the results is in rheometers. Here a
uniform shear rate is assumed to be present in the gap when measuring the
torque needed to rotate the plate or cylinder - depending on actual rheometer
type. With shear banding present in the flow, the conditions considered for the
measurements are obviously violated inducing errors in the rheology measure-
ments. When modeling the flow shear banding is not normally accounted for.
In the analytical model presented in this paper the shear rate is assumed to be
homogeneously distributed across the channel; in the plotted velocity profiles
in Figs. 6-7 this is showed by the continuously varying velocity in the viscous

Grease flow in an elbow channel 11
layer between the channel wall and the position of the plug region if present,
otherwise the velocity peak in the channel. Shear banding may be responsible
for the deviation between the analytical model and the µPIV measurements
viewed in Fig. 7. It is close at hand to expect that the pressure can cause the
deviation with respect to what is discussed in the previous subsection, but the
pressure affect the velocity continuously from the location of the plug region
as viewed in Fig. 6. From Fig. 7a it follows that in the viscous layer in connec-
tion to the inner radius (located approximately in the region 0 < r∗<0.4) the
general trend is a parabolical velocity profile as described by the analytical
solution. The deviation from the flow represented by the velocity u∗≈0.3 at
r∗≈0.15 indicates a local effect. A similar behavior is seen for the higher flow
rate case in Fig. 7c. Here the velocity profiles in connection to the outer radius
show a more continuous concave shape; this effect is coupled to the geometry
where the concave outer wall induce a more continuous flow compared to the
convex inner radius which resembles the geometry of a rounded backward fac-
ing step. Lerouge at al. [16] (Fig. III.1, p.19) show that shear banding due to
the discontinuous shear rate distribution in the flow is characterized by notches
in the graph when plotting the flow velocity gradient versus the velocity. The
notches are the locations of the critical shear rates in the flow. In a shear
stress/shear rate plot the shear banding effect is viewed as an unstable region
within the limits formed by the critical shear rates. With this background it
can be argued that the location of the critical shear rates in Fig. 7a are at
r∗≈0.15 and r∗≈0.25, and r∗≈0.8 and r∗≈0.9 at respective viscous
layers. No general conclusion can be drawn regarding the shear banding effect
related to the NLGI grade of the grease. In terms of the flow rate the tendency
is that the effect we refer to as shear banding in this section, is reduced for
higher flow rates. This result can be related to the origin of shear banding
illustrated by Lerouge et al. which can be applicable to greases considering a
similarity between the grease thickener and giant micelles: For high shear rates
the thickener fibres are aligned with the flow while as the shear rate decreases
the fibre orientation becomes more random causing a varying rheology in the
grease. Consequently, if the flow rate is high enough the shear rate in the flow
is high enough to form aligned fibres throughout the whole flow domain and
hence the shear banding effect will vanish.
4.3 Wall slip and flow evolution in the elbow channel
Modeling channel flow, or flow in the gap in a rheometer, a no slip boundary
condition is typically applied. In grease flow this condition is showed to easily
be not fulfilled [29, 7, 6]. The origins of wall slip are not entirely understood;
that phase separation is responsible for the slip in connection to the solid
boundary most researchers agree on. Wall slip effects through a thin layer at
the walls consisting of a less viscous (i.e. lower consistency index) grease than
the bulk grease [6,23]. The lower consistency is by Barnes [3] suggested to
be due to oil separation between the base oil and the grease thickener agent.

12 L.G. Westerberg et al.
The concept of wall slip can be divided into two categories: i) true slip, where
there is a discontinuity in the velocity field at the fluid-solid interface, and
ii) apparent slip, where there is an inhomogeneous thin layer of a fluid adjacent
to the wall with different rheological properties to the bulk of fluid enabling
fluid movement [27]. Li et al. [19] also showed that in a free-surface flow oil
bleeding is present where the grease is sheared. This result supports the idea of
oil bleeding being responsible for wall slip as oil bleeding in connection to the
walls in closed (channel) flows in turn is supported by the high shear stresses
close to the solid boundaries.
Fig. 7 shows that the wall slip effect is small in the elbow, both for the soft
NLGI 00-grease and the thick NLGI 2-grease. The wall slip effect in the flow
is characterized by u∗=u∗
slip >0 when the velocity profile is extrapolated
towards r∗= 0. The resolution of the flow is finite and set by the microscope
objective magnification, meaning the first measured velocity is somewhere be-
tween the solid wall and the distance set by the length between the data points
in the velocity grid. As presented in §3 the micro-channel is built of a polymeric
material (VeroWhitePlus). The channel surface roughness has an Ra value of
1.85 µm with standard deviation of 0.78 µm which should be compared to
the channel Ra value in the work by Westerberg et al. [29] lying in the range
0.1-5 µm. In that study it was concluded that these Ra values did not have
any influence on the wall slip observed in the straight channel used. Czarny [7]
on the other hand showed that the surface roughness has impact on the wall
slip, but this yields for the case of a surface roughness of several orders of
magnitude larger than considered by Westerberg et al who used bearing steels
as reference. Westerberg et al. also showed that the slip effect is not depen-
dent on the actual surface material which in their study comprised steel (AISI
52100), polyamide and brass. Another conclusion was that the slip decreases
with increasing flow rate. For elbow channel flow the shear stress is higher in
the elbow than in the straight sections (inlet and outlet) as the curvature of
the geometry is included in the expression for the shear stress; see Eq. 10. Con-
sidering a constant velocity in the ϕ-direction the shear stress is proportional
to the inverse of the radial position r. Having negligible wall slip in the elbow
section can hence be connected to the conclusion of a reduced wall slip with
increasing shear forces. Czarny [7] also showed that wall slip was accentuated
for low shear rates. In Fig. 8 velocity profiles at the inlet, elbow and outlet
are compared. Considering subfigure (a) showing the low flow rate case for
the NLGI 00-grease; if the velocity profile for the inlet and outlet regions in
the channel (square and cross labeled curves) are extrapolated towards the
inner- and outer boundary respectively a slip velocity of approximately 0.3
is obtained, while the corresponding slip velocity at the elbow (dotted curve)
indicates a zero slip velocity. In (c) we see that that for a high flow rate the
velocity profiles at all three locations have the same parabolical form. For the
thick NLGI 2-grease ((b) and (d)) the difference in slip behavior is not as dis-
tinct; a clear observation is though that the length of the plug region is reduced
at the elbow. The results by Westerberg et al. [29] also support this since they
showed that the plug region decreases when the pressure drop (flow rate) is

Grease flow in an elbow channel 13
increased. An interesting observation when comparing the low- and high flow
rates for the NLGI 2 grease is that the velocity profile in the elbow not is
affected by the higher flow rate, indicating that the shear rate in the elbow
even at low flow rates induces a fully yielded grease. These results agree with
the conclusions made by Tamayol et al. [28] stating that the slip increases with
the Knudsen number (Kn =λ/L i.e. [mean free path]/[characteristic length
scale]): with an increasing consistency (which we can treat as a viscosity for
the non-Newtonian grease) the mean free path increases and consequently Kn
increases [26].
4.4 Stick-slip type of motion during transitory period
Another interesting observation when comparing the velocity profiles during
increased flow rates was a variation in the maximum speed in the channel for
a constant feeding flow rate. This behavior is strange considering the grease
being a continuum flowing medium; with the same flow rate and constant
channel cross section the maximum speed of the flow should remain constant.
Based on these observations, the evolution of the maximum speed at the inlet
during the transitory period has been investigated; see Fig. 9. Starting from
rest the specified feeding flow rate is applied until stationary conditions are
reached. A high- and low flow rate are considered. Comparing subfigure a and
b it follows that stationary conditions are reached fast for the high flow rate
case, while stationary conditions for the low flow rate are never reached; the
speed shows a great variation as the pressure is built up in the system with
a behavior similar to a stick-slip motion. An explanation of this result can be
coupled to the non-homogeneous properties of the grease also shown as shear
bands in the flow. As the pressure is built up in the grease the shear forces in
the grease are spatially varying due to a varying composition of thickener and
base oil which in turn induces varying rheological properties of the grease. This
also means that the pressure distribution in the grease is varying. Once the
shear stress (locally) in the grease exceeds the yield stress, the flow is released
and the locally accumulated pressure introduces a high pressure gradient com-
pared to the main driving pressure induced by the syringe pump, causing the
speed of the grease flow to accelerate - which in Fig. 9a is shown as the rapid
increases (peaks) in the measured speed. Once this transient effect is decaying
the shear stress in the grease will approach the yield stress once again causing
a retardation of the motion of the grease. This effect is visualized by the sud-
den drop in speed shown in Fig. 9a. For the case of a high flow rate the shear
forces in the grease exceeds the yield stress throughout the whole domain,
causing a continuous deformation of the grease and a continuous flow in the
channel. The coupling of the observed phenomena to the yield stress behavior
of the grease is observed when comparing the behavior of the NLGI 00- and
NLGI 2-grease: for the NLGI 2-grease flow at high flow rates the transition
from the onset of grease motion to stationary flow occur on a short time scale,
while the NLGI 00-grease due to its negligible yield stress start to move when

14 L.G. Westerberg et al.
the feeding pressure is set and hence the corresponding transition time from
the onset of grease motion to a stationary speed is longer. The time needed to
achieve a stationary speed calculated from the onset of the driving pressure
is however equal for both greases. The measured stick-slip type of behavior
is interesting on a fundamental level with respect to the grease rheology and
its coupling to the grease flow, but also with respect to applications such as
pumping/feeding of grease - e.g. in lubrication systems discussed in §4.2. With
a too low feeding pressure the flow rate of grease into the actual component
will vary significantly, which in turn may have negative consequences on the
desired lubrication due to too much grease or practically no inflow of grease.
This scenario becomes accentuated at low temperatures as the grease viscosity
and yield stress change dramatically.
4.5 Grease flow in pipes used in lubrication systems
The observed flow characteristics presented above in elbow geometries, may
be of certain interest for lubrication systems. The present work does not cover
research on lubrication systems as such, but still the results may have value for
this engineering application as elbow- and t-joint fittings are occurring in these
systems. Problems with disruptions in the feeding of grease through the pipes
and fittings are not uncommon (Victoria Van Camp, Vice President Product
Management and R&D at SKF Lubrication Systems, St Louis, US. Private
communication May 23, 2014). Lubrication systems have an important role in
lubrication as the right amount of grease is supposed to be delivered to the
right location at the right time. A challenge is to find a grease whose properties
satisfies both the lubrications systems’- and the bearing’s requirements. And
here knowledge of the flow behaviour observed in this paper is of interest.
From the model observations it follows that the circulation in the elbow
due to flow separation increases with the driving pressure. The condition for
when circulation in the elbow is initiated is also strongly dependent on the con-
sistency index (Kin the Herschel-Bulkley rheology model) and on the shear
thinning parameter n(<1). This result seems natural as a stiffer grease due
to lower bulk inertial forces and higher viscosity inherits a resistance to sepa-
rate from the elbow walls. For a decreasing n-value the viscosity of the grease
will be lower for the same shear rate, meaning that circulation in the elbow
theoretically could be initiated, or ceased, for constant pressure by changing
the shear thinning properties of the grease: increase the n-value to prevent
circulation, or decrease the n-value to initiate circulation. The circulation in
the elbow is initiated by the shear in the flow induced by the curved elbow
geometry; the flow in the straight section(s) does however not experience the
same magnitude of shear stresses meaning the well-ordered plug-, or parabol-
ical upstream of the elbow enter a region with a swirling flow. This swirl is
likely to act as an obstacle for the approaching flow, or in worst case block
the entire flow. Other effects such as phase separation (oil bleeding) due to
the shear stresses in the elbow may contribute to block the elbow fitting. So

Grease flow in an elbow channel 15
in practical terms, based on these results blocked pipes could be avoided by
decreasing the feeding pressure, increase the curvature radius of the fitting
parts of the pipes, and/or have a grease with less pronounced shear thinning
properties.
The stick-slip type of motion observed when investigating the speed in the
channel (§4.4) also is interesting for the flow in lubrication systems. For low
feeding flow rates where the shear stress in the bulk grease balances on the
yield stress value, the pressure is locally increased which in turn increases the
shear stress until the yield condition is passed and a rapid increase in velocity
is obtained. With the released flow the shear stress magnitude distribution
in the flow decreases, implying the grease will approach the yield threshold
condition once again. These variations in the flow will contribute to challenges
in terms of keeping a uniform flow rate into the system where the lubricated
contact surfaces are located. The pumpability of grease through the pipes in
lubrication systems will also be strongly affected by low temperatures as the
grease stop bleeding oil and its flow properties will be different due to a stiffer
(i.e. more viscous) grease ([20], p.19).
Wall slip is an interesting phenomena: as presented in §4.3 there is no
consensus on whether the slip is caused by a large concentration gradient of
base oil or thickener. If we as a working hypothesis consider the former to
be the primary reason for the existence of wall slip in grease flow, it is not
unlikely that wall slip may be of value for the pumpability as slip reduces the
friction and consequently a lower pressure is needed.
5 Summary and concluding remarks
In this paper the flow of lubricating greases have been analyzed in an el-
bow channel with a rectangular cross section of 250x1000 µm. The problem
has been modeled and validated with velocity profiles from flow visualiza-
tions using micro Particle Image Velocimetry.. The analytical model is built
from the fluid equation of motion considering the Herschel-Bulkley rheology
model. Three Lithium-based greases with NLGI grade 00, 1 and 2 respectively
have been used and the flow has been analyzed for flow rates in the range
0.002 ml/min to 0.5 ml/min. It is shown that there is a good match between
the analytical model of the flow in the elbow and the measured velocity pro-
files. The analytical solution for the velocity across the elbow section is shown
to be sensitive to the pressure in the angular direction (ϕ, considering cylin-
drical coordinates). As the pressure increases the velocity profile gradually
approaches a reversal indicating that flow separation occurs in the elbow in
connection to the solid boundaries. This result has a direct application in lu-
brication systems where an effective feed of grease from the reservoir to the
component to be lubricated is highly important. With separation present in
the flow, the possibility to effectively pump the grease through the pipe sys-
tem is disturbed. Another finding with direct relevance to lubrication systems
is that for low flow rates the grease experiences large velocity fluctuations in

16 L.G. Westerberg et al.
the channel, i.e. a stick-slip type of motion. This effect disappears for high
flow rates, concluding that in order to achieve an even flow rate through (for
instance) a lubrication system the applied flow rate itself cannot be too low.
Flow visualizations show that wall slip is present and it is found to be reduced
in the elbow compared to the in- and outlet sections of the channel. The con-
clusion is that this effect - which mainly is observed for low flow rates and for
the thinnest grease with NLGI grade 00, is caused by the higher shear mag-
nitude in the elbow flow caused by the curvature. For high flow rates the slip
effect decreases and the overall flow is unaffected by the elbow. Shear band-
ing due to a non-uniform distribution of the shear rate in the channel is also
observed.
Acknowledgements The authors would like to thank Dr. Torb j¨orn Green and Dr. Henrik
Lycksam at the Division of Fluid and Experimental Mechanics, LTU for their help with the
µPIV measurements. This project has been funded by the Swedish Research Council (VR)
and by KIC InnoEnergy.
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18 L.G. Westerberg et al.
Table 1 Rheology data for the used NLGI 00, 1 and 2 greases based on the Herschel-Bulkley
rheology model.
τ0[Pa] K[Pa ·sn]n[-]
NLGI 00 1 1.85 1
NLGI 1 189 4.1 0.80
NLGI 2 500 8.2 0.63
Flow direction
r
z
׋
Inner radius Outer
radius
Fig. 1 Schematic drawing of the elbow channel geometries used and the cylindrical coor-
dinate system to describe the flow in the elbow section. The regions marked with dashed
lines are the ones where the velocity profiles are measured at the inlet, elbow and outlet
respectively.
Plug region
Slip layers
Viscous
regions
uࢥ(r)
r
r1rp1
u1
up1
u2
up2
r2
rp2
uࢥ1(r)
uࢥ2(r)
Fig. 2 Schematic view of a typical velocity profile in the channel elbow with a plug region
present. Not to scale. The dots represents the measured velocity from the µPIV experiments.
r1,2, rp1,p2and u1,2, up1,p2are the locations and velocity of the first measured velocity
value and the location of the yield point respectively in connection to respective channel
boundary.

Grease flow in an elbow channel 19
(a)
(a)
Outlet
Inlet Micro-channel
Centering columns
Outlet
Sealing
(a) (b)
(b)
Fig. 3 Dimensions of the micro-channel (a) and an overview of the channel setup (b). Top
view (left) and bottom view (right).

20 L.G. Westerberg et al.
Fig. 4 Fabrication steps for the micro-channel.

Grease flow in an elbow channel 21
Microscope
High speed camera
Micro channel
Syringe pump
Fiber cable from
laser
Manometer
(a)
(b)
Fig. 5 Setup of the µPIV experiment (a) and the principles of PIV (b). Subfigure (b) is
courtesy of Dantec Dynamics A/S.

22 L.G. Westerberg et al.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
(a) α=−100.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
(b) α=−200.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
(c) α=−500.
0 0.2 0.4 0.6 0.8 1
−0.5
0
0.5
1
(d) α=−1000.
Fig. 6 Dimensionless velocity profiles at the elbow for the NLGI1 grease with a flow rate
of 0.02 ml/min. Channel with cross-section of 250x1000 µm. As characteristic velocity and
length scale the maximum velocity and the channel width (250 µm) have been used. Dot-
ted:µPIV measurements, dash dotted: analytical model. On the y-axis is the dimensionless
velocity (u∗(r∗)) and on the x-axis is the distance from the inner radius (r∗). αis the
pressure parameter in the analytical model introduced in Eq. (6). Figs. a-d shows the evo-
lution of the analytical solution as the pressure in the ϕ-direction increases. The values
of the rheological parameters in the Herschel-Bulkley rheology model are for these plots:
n= 1 [−], K = 7 [Pa ·sn] and τ0= 160 [Pa].

Grease flow in an elbow channel 23
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
(a) NLGI00 2 ·10−2ml/min
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
(b) NLGI2 2 ·10−2ml/min
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
(c) NLGI00 0.5 ml/min
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
(d) NLGI2 0.5 ml/min
Fig. 7 Velocity profiles in the channel elbow for the NLGI00- and NLGI2 grease and for
two different flow rates. Dotted:µPIV measurements, dash dotted: analytical model. Di-
mensionless quantities using a characteristic length scale of 250 µm (channel width) and a
characteristic velocity equal to the maximum velocity. On the y-axis is the velocity and on
the x-axis is the distance from the inner radius. Upper row: flow rate 0.02 ml/min, bottom
row: flow rate 0.5 ml/min.

24 L.G. Westerberg et al.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
(a) NLGI00 2 ·10−3ml/min
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
(b) NLGI2 2 ·10−3ml/min
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
(c) NLGI00 0.5 ml/min
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
(d) NLGI2 0.5 ml/min
Fig. 8 Measured velocity profiles at the inlet (square), elbow (dotted), and outlet (cross).
Dimensionless quantities using a characteristic length scale of 250 µm (channel width) and
a characteristic velocity equal to the maximum velocity. On the y-axis is the velocity and on
the x-axis is the distance from the inner radius. Upper row: low flow rate (0.002 ml/min),
bottom row: high flow rate (0.5 ml/min).

Grease flow in an elbow channel 25
0,0E+00
5,0E-05
1,0E-04
1,5E-04
2,0E-04
2,5E-04
0:00 2:24 4:48 7:12 9:36 12:00
NLGI02 NLGI00 Vt=0.0001333
5,00E-02
(a) Low flow rate (0
0,00E+00
1,00E-02
2,00E-02
3,00E-02
4,00E-02
5,00E-02
0:00 0:02 0:05 0:08 0:11 0:14 0:17 0:20 0:23
NLGI2 NLGI00 Vt=0.033
(b) High flow rate
Fig. 9 Speed evolution of the grease flow during the transitory period measured at a control
point in the straight section upstream of the elbow (cf. Fig. 1) for the three greases used.
On the y-axis is the speed in m/s and on the x-axis the elapsed time (hh:mm). Vtis the
theoretical speed considering the flow rate and cross section area of the channel assuming a
uniform velocity distribution.